Intersection of Symmetric Relations is Symmetric

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Theorem

The intersection of two symmetric relations is also a symmetric relation.


Proof

Let $\mathcal R_1$ and $\mathcal R_2$ be symmetric relations on a set $S$.

Let $\mathcal R_3 = \mathcal R_1 \cap \mathcal R_2$.

Then:

\(\displaystyle \) \(\) \(\displaystyle \left({x, y}\right) \in \mathcal R_3\)
\(\displaystyle \) \(\implies\) \(\displaystyle \left({x, y}\right) \in \mathcal R_1 \cap \mathcal R_2\) by definition of $\mathcal R_3$
\(\displaystyle \) \(\implies\) \(\displaystyle \left({x, y}\right) \in \mathcal R_1 \land \left({x, y}\right) \in \mathcal R_2\) Definition of Intersection
\(\displaystyle \) \(\implies\) \(\displaystyle \left({y, x}\right) \in \mathcal R_1 \land \left({y, x}\right) \in \mathcal R_2\) $\mathcal R_1$ and $\mathcal R_2$ are symmetric
\(\displaystyle \) \(\implies\) \(\displaystyle \left({y, x}\right) \in \mathcal R_1 \cap \mathcal R_2\) Definition of Intersection
\(\displaystyle \) \(\implies\) \(\displaystyle \left({y, x}\right) \in \mathcal R_3\) by definition of $\mathcal R_3$

$\blacksquare$